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Extra resources for Approximate Analytical Methods for Solving Ordinary Differential Equations
Are in terms of a1. We know that a0 ↑ 0 . Let us find the value of a1. 15), which gives a1 = 0 . 30 A p p r ox im at e A n a ly ti c a l M e t h o d s Now, using the recurrence relation given previously in this example, we obtain a0 3! 4 5! a2 = − and so on. Thus, the first Frobenius solution is y1 ( x ) = a0 x 1/2 1 − x2 x4 + − . 2! 5! Now, y 2 = a 0* x 1/2 1 − x2 x4 + − 2! 5! ∫ 1 x 1/2 x2 x4 1− + − 2! 5! 2 e − 1 ∫ x dx After straightforward but lengthy calculations, we obtain y 2 = a 0* x −1/2 1 − x2 x4 + − 2!
Otherwise, x0 is termed an IRSP. For instance, (i) x = 0 is an RSP of y − cos x y = 0. x Note that P (x ) = − cos x 1 x x3 =− + − + x x 2 24 24 A p p r ox im at e A n a ly ti c a l M e t h o d s (ii) x = 1 is an IRSP of y − 1 y = 0. 11) if ( x − x0 ) P ( x ) is analytic at x0 . At RSPs, we look for solutions of the form ∞ y= ∑ a (x − x ) n n=0 0 n +m , a0 ≠ 0 because if a0 = 0 , then some positive integral power of x can be factored out of the power series part and can be combined with ( x − x0 )n .
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